Route to thermalization in the α-Fermi–Pasta–Ulam system

We study the original α-Fermi–Pasta–Ulam (FPU) system with N = 16, 32, and 64 masses connected by a nonlinear quadratic spring. Our approach is based on resonant wave–wave interaction theory; i.e., we assume that, in the weakly nonlinear regime (the one in which Fermi was originally interested), the large time dynamics is ruled by exact resonances. After a detailed analysis of the α-FPU equation of motion, we find that the first nontrivial resonances correspond to six-wave interactions. Those are precisely the interactions responsible for the thermalization of the energy in the spectrum. We predict that, for small-amplitude random waves, the timescale of such interactions is extremely large and it is of the order of 1/ϵ8, where ϵ is the small parameter in the system. The wave–wave interaction theory is not based on any threshold: Equipartition is predicted for arbitrary small nonlinearity. Our results are supported by extensive numerical simulations. A key role in our finding is played by the Umklapp (flip-over) resonant interactions, typical of discrete systems. The thermodynamic limit is also briefly discussed.

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Nonlinear and three-wave resonant interactions in magnetohydrodynamics

Journal of Plasma Physics ◽

10.1017/s0022377800008370 ◽

2000 ◽

Vol 63 (5) ◽

pp. 393-445 ◽

Cited By ~ 9

Author(s):

G. M. WEBB ◽

A. R. ZAKHARIAN ◽

M. BRIO ◽

G. P. ZANK

Keyword(s):

Wave Field ◽

Wave Interaction ◽

Resonant Interaction ◽

Alfven Wave ◽

Magnetoacoustic Wave ◽

Wave Interactions ◽

Weakly Nonlinear ◽

Resonant Interactions ◽

Resonant Triads ◽

High Frequency Waves

Hamiltonian and variational formulations of equations describing weakly nonlinear magnetohydrodynamic (MHD) wave interactions in one Cartesian space dimension are discussed. For wave propagation in uniform media, the wave interactions of interest consist of (a) three-wave resonant interactions in which high-frequency waves may evolve on long space and time scales if the wave phases satisfy the resonance conditions; (b) Burgers self-wave steepening for the magnetoacoustic waves, and (c) mean wave field effects, in which a particular wave interacts with the mean wave field of the other waves. The equations describe four types of resonant triads: slow–fast magnetoacoustic wave interaction, Alfvén–entropy wave interaction, Alfvén–magnetoacoustic wave interaction, and magnetoacoustic–entropy wave interaction. The formalism is restricted to coherent wave interactions. The equations are used to investigate the Alfvén-wave decay instability in which a large-amplitude forward propagating Alfvén wave decays owing to three-wave resonant interaction with a backward-propagating Alfvén wave and a forward-propagating slow magnetoacoustic wave. Exact solutions of the equations for Alfvén–entropy wave interactions are also discussed.

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Resonant wave interactions in a stratified shear flow

Journal of Fluid Mechanics ◽

10.1017/s0022112088001351 ◽

1988 ◽

Vol 190 ◽

pp. 357-374 ◽

Cited By ~ 11

Author(s):

R. Grimshaw

Keyword(s):

Shear Flow ◽

Gravity Waves ◽

Critical Level ◽

Internal Gravity Waves ◽

Wave Interactions ◽

Resonant Wave ◽

Resonant Interactions ◽

Stratified Shear Flow ◽

Weak Resonance ◽

Background Shear

Resonant interactions between triads of internal gravity waves propagating in a shear flow are considered for the case when the stratification and the background shear flow vary slowly with respect to typical wavelengths. If ωn, kn(n = 1, 2, 3) are the local frequencies and wavenumbers respectively then the resonance conditions are that ω1 + ω2 + ω3 = 0 and k1 + k2 + k3 = 0. If the medium is only weakly inhomogeneous, then there is a strong resonance and to leading order the resonance conditions are satisfied globally. The equations governing the wave amplitudes are then well known, and have been extensively discussed in the literature. However, if the medium is strongly inhomogeneous, then there is a weak resonance and the resonance conditions can only be satisfied locally on certain space-time resonance surfaces. The equations governing the wave amplitudes in this case are derived, and discussed briefly. Then the results are applied to a study of the hierarchy of wave interactions which can occur near a critical level, with the aim of determining to what extent a critical layer can reflect wave energy.

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Linear and Weakly Nonlinear Energetics of Global Nonhydrostatic Normal Modes

Journal of the Atmospheric Sciences ◽

10.1175/jas-d-19-0131.1 ◽

2019 ◽

Vol 76 (12) ◽

pp. 3831-3846 ◽

Cited By ~ 1

Author(s):

Carlos F. M. Raupp ◽

André S. W. Teruya ◽

Pedro L. Silva Dias

Keyword(s):

Normal Modes ◽

Physical Space ◽

Wave Mode ◽

Finite Amplitude ◽

Vertical Acceleration ◽

Wave Interactions ◽

Weakly Nonlinear ◽

Resonant Wave ◽

Zonal Wavenumber ◽

Resonant Triads

Abstract Here the theory of global nonhydrostatic normal modes has been further developed with the analysis of both linear and weakly nonlinear energetics of inertia–acoustic (IA) and inertia–gravity (IG) modes. These energetics are analyzed in the context of a shallow global nonhydrostatic model governing finite-amplitude perturbations around a resting, hydrostatic, and isothermal background state. For the linear case, the energy as a function of the zonal wavenumber of the IA and IG modes is analyzed, and the nonhydrostatic effect of vertical acceleration on the IG waves is highlighted. For the nonlinear energetics analysis, the reduced equations of a single resonant wave triad interaction are obtained by using a pseudoenergy orthogonality relation. Integration of the triad equations for a resonance involving a short harmonic of an IG wave, a planetary-scale IA mode, and a short IA wave mode shows that an IG mode can allow two IA modes to exchange energy in specific resonant triads. These wave interactions can yield significant modulations in the dynamical fields associated with the physical-space solution with periods varying from a daily time scale to almost a month long.

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A variational method for weak resonant wave interactions

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1969.0056 ◽

1969 ◽

Vol 309 (1499) ◽

pp. 551-577 ◽

Cited By ~ 79

Keyword(s):

Variational Method ◽

Internal Gravity Waves ◽

Wave Number ◽

Local Conservation ◽

Wave Interactions ◽

First Order ◽

Resonant Wave ◽

Resonant Interactions ◽

Energy And Momentum ◽

Phase Angles

Whitham’s variational method is formulated so as to apply to weak second-order resonant interactions among waves whose amplitudes and phase angles vary slowly with position and time. The method is applied in detail to capillary-gravity wave interactions. An internal gravity waves problem is also discussed briefly. The method leads to new and substantial simplifications of the interaction equations. This makes possible the proof of local conservation of total mean wave energy and momentum laws. These, together with another integral of the motion, are found to be of central importance in classifying and characterizing the slow modulations of planewave-like form. Such a classification is given in detail for all initial values of phase angles and relative amplitudes. All progressive uniform waves in the capillary range are found to be unstable with perturbation growth rates which can be of first order in the wave slopes. In this formulation amplitude dependent first-order corrections of classical frequency and/or wave-number arise for all waves participating in a resonance. A few predictions which could be verified by simple experiments are made.

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An analytical model of the evolution of a Stokes wave and its two Benjamin–Feir sidebands on nonuniform unidirectional current

Nonlinear Processes in Geophysics ◽

10.5194/npg-22-313-2015 ◽

2015 ◽

Vol 22 (3) ◽

pp. 313-324 ◽

Cited By ~ 2

Author(s):

I. V. Shugan ◽

H. H. Hwung ◽

R. Y. Yang

Keyword(s):

Schrödinger Equation ◽

Spatial Scale ◽

Schrodinger Equation ◽

Wave Interaction ◽

Experimental Results ◽

Stokes Wave ◽

Initial State ◽

Weakly Nonlinear ◽

Resonant Interactions ◽

Cubic Schrödinger Equation

Abstract. An analytical weakly nonlinear model of the Benjamin–Feir instability of a Stokes wave on nonuniform unidirectional current is presented. The model describes evolution of a Stokes wave and its two main sidebands propagating on a slowly varying steady current. In contrast to the models based on versions of the cubic Schrödinger equation, the current variations could be strong, which allows us to examine the blockage and consider substantial variations of the wave numbers and frequencies of interacting waves. The spatial scale of the current variation is assumed to have the same order as the spatial scale of the Benjamin–Feir (BF) instability. The model includes wave action conservation law and nonlinear dispersion relation for each of the wave's triad. The effect of nonuniform current, apart from linear transformation, is in the detuning of the resonant interactions, which strongly affects the nonlinear evolution of the system. The modulation instability of Stokes waves in nonuniform moving media has special properties. Interaction with countercurrent accelerates the growth of sideband modes on a short spatial scale. An increase in initial wave steepness intensifies the wave energy exchange accompanied by wave breaking dissipation, resulting in asymmetry of sideband modes and a frequency downshift with an energy transfer jump to the lower sideband mode, and depresses the higher sideband and carrier wave. Nonlinear waves may even overpass the blocking barrier produced by strong adverse current. The frequency downshift of the energy peak is permanent and the system does not revert to its initial state. We find reasonable correspondence between the results of model simulations and available experimental results for wave interaction with blocking opposing current. Large transient or freak waves with amplitude and steepness several times those of normal waves may form during temporal nonlinear focusing of the waves accompanied by energy income from sufficiently strong opposing current. We employ the model for the estimation of the maximum amplification of wave amplitudes as a function of opposing current value and compare the result obtained with recently published experimental results and modeling results obtained with the nonlinear Schrödinger equation.

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Theoretical and experimental studies of gravity wave interactions

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1967.0125 ◽

1967 ◽

Vol 299 (1456) ◽

pp. 104-119 ◽

Cited By ~ 21

Keyword(s):

Gravity Waves ◽

Experimental Studies ◽

Internal Gravity Waves ◽

Stokes Wave ◽

Wave Interactions ◽

Principal Characteristics ◽

Resonant Wave ◽

Resonant Interactions ◽

Interaction Distance ◽

Wave Modes

This paper is concerned with various aspects of the resonant interactions among waves. An experiment was suggested by Longuet-Higgins (1962) to detect this type of interaction among surface waves. This was subsequently performed by Longuet-Higgins & Smith (1966) and by McGoldrick, Phillips, Huang & Hodgson (1966). The results of the two sets of experiments are compared. Together they demonstrate very clearly the principal characteristics of the interaction; the maximum response at resonance and the linear growth with interaction distance, the decrease in band width with interaction distance and the shift of the resonance point that results from the amplitude dispersion. It is shown further that the instability of the Stokes wave, discovered and analysed by Benjamin & Feir, can be described in terms of these interactions and that it is not restricted to purely two dimensional motion. A Stokes wave is unstable to a disturbance containing a pair of wavenumbers defined by any point in the zone just inside the figure-of-eight loop shown in figure 12. Another example of resonant wave interactions is provided by short, internal gravity waves in a stratified fluid with constant Brunt-Väisälä frequency. The interactions among Fourier modes are considered, and it is shown that there arise both free and forced modes. In the latter, the dispersion relation for internal waves is not satisfied; there is no particular relation between wavenumber and frequency. The amplitudes of these are small compared with those of the internal wave modes provided the harmonic mean of the vorticity in the two interacting waves is small compared with the Brunt-Väisälä frequency. The motion then consists of interacting internal gravity waves, whose interaction sets are closed. On the other hand, if the forced components are comparable in magnitude with the wave modes, these interact strongly and indiscriminately; a ‘cascade’, characteristic of turbulence, develops.

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Observation of resonant interactions among surface gravity waves

Journal of Fluid Mechanics ◽

10.1017/jfm.2016.576 ◽

2016 ◽

Vol 805 ◽

Cited By ~ 20

Author(s):

F. Bonnefoy ◽

F. Haudin ◽

G. Michel ◽

B. Semin ◽

T. Humbert ◽

...

Keyword(s):

Gravity Waves ◽

Phase Locking ◽

Wave Interaction ◽

Quantitative Agreement ◽

Acute Angle ◽

Experimental Results ◽

Surface Gravity ◽

Surface Gravity Waves ◽

Resonant Wave ◽

Resonant Interactions

We experimentally study resonant interactions of oblique surface gravity waves in a large basin. Our results strongly extend previous experimental results performed mainly for perpendicular or collinear wave trains. We generate two oblique waves crossing at an acute angle, while we control their frequency ratio, steepnesses and directions. These mother waves mutually interact and give birth to a resonant wave whose properties (growth rate, resonant response curve and phase locking) are fully characterized. All our experimental results are found in good quantitative agreement with four-wave interaction theory with no fitting parameter. Off-resonance experiments are also reported and the relevant theoretical analysis is conducted and validated.

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On internal wave–shear flow resonance in shallow water

Journal of Fluid Mechanics ◽

10.1017/s0022112097007593 ◽

1998 ◽

Vol 354 ◽

pp. 209-237 ◽

Cited By ~ 7

Author(s):

VYACHESLAV V. VORONOVICH ◽

DMITRY E. PELINOVSKY ◽

VICTOR I. SHRIRA

Keyword(s):

Shear Flow ◽

Internal Wave ◽

Internal Waves ◽

Solitary Waves ◽

Evolution Equations ◽

Wave Interaction ◽

Surface Velocity ◽

Wave Flow ◽

Weakly Nonlinear ◽

Resonant Wave

The work is concerned with long nonlinear internal waves interacting with a shear flow localized near the sea surface. The study is focused on the most intense resonant interaction occurring when the phase velocity of internal waves matches the flow velocity at the surface. The perturbations of the shear flow are considered as ‘vorticity waves’, which enables us to treat the wave–flow resonance as the resonant wave–wave interaction between an internal gravity mode and the vorticity mode. Within the weakly nonlinear long-wave approximation a system of evolution equations governing the nonlinear dynamics of the waves in resonance is derived and an asymptotic solution to the basic equations is constructed. At resonance the nonlinearity of the internal wave dynamics is due to the interaction with the vorticity mode, while the wave's own nonlinearity proves to be negligible. The equations derived are found to possess solitary wave solutions of different polarities propagating slightly faster or slower than the surface velocity of the shear flow. The amplitudes of the ‘fast’ solitary waves are limited from above; the crest of the limiting wave forms a sharp corner. The solitary waves of amplitude smaller than a certain threshold are shown to be stable; ‘subcritical’ localized pulses tend to such solutions. The localized pulses of amplitude exceeding this threshold form infinite slopes in finite time, which indicates wave breaking.

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